One-dimensional Fermi gases: Density-matrix renormalization group study of ground state properties and dynamics

Transcription

1 Renormalization Group Approach from Ultra Cold Atoms to the Hot QGP Yukawa Institute for Theoretical Physics, Kyoto, Japan One-dimensional Fermi gases: Density-matrix renormalization group study of ground state properties and dynamics 30 August 2011 Masaki TEZUKA (Kyoto University)

2 Plan of the talk Introduction Low-dimensional cold atom systems Density-matrix renormalization group Harmonically trapped imbalanced system Larkin-Ovchinnikov state Optical lattice with disorder Proving superfluidity by dynamics Collision dynamics on-classical reflectance Summary PRL 100, (2008) ew J. Phys 12, (2010) PRA 82, (2010) arxiv:

3 Collaborators Masahito Ueda (University of Tokyo) Harmonically trapped imbalanced system Antonio M. García-García (Cambridge University) Optical lattice with disorder PRA 82, (2010) PRL 100, (2008) ew J. Phys 12, (2010) Jun ichi Ozaki (Kyoto University) orio Kawakami (Kyoto University) Collision dynamics arxiv:

4 Low-dimensional cold atom systems single atom detection 2D hologram 2D Bose-Einstein condensates (BEC) released and collide Berezinskii-Kosterlitz-Thouless (BKT) transition is observed Hadzibabic et al.: ature 441, 1118 (2006) WS Bakr et al.: ature 462, 74 (2009)

5 Low-dimensional cold atom systems Quasi-1D system in optical lattices 2D lattice of 1D systems 1D Bosons: absence of thermalization Kinoshita et al.: ature 440, 900 (2006) see e.g. I. Bloch, ature Phys. (2005) Speckle potential for bosons Billy et al.: ature 453, 891 (2008) 1D : Strong radial confinement Radial level separation >> energy scale of axial motion (Fermi energy in fermionic system) Low-dimensional but not necessarily solvable How to simulate numerically without uncontrollable approximations?

6 Density-Matrix Renormalization Group System S = B + s + s + B ijj' i' cijj' i' i j j' i' kk ' ijj' i' kj' i' kj' i' i' j' i' j' kj' i' (DMRG) Partial density matrix for the left block L (k (ij)) c c * k ' j' i' Has all information on L = B + s when S is in the target state Ψ> environment block Diagonalize ρ : states with larger eigenvalues are more important cf. In RG (numerical renormalization group) the lowest energy states are kept Reduce the dimension to m of the Hilbert space for L = renormalization S. R. White: PRL (1992) PRB (1993) Reviews: Schollwöck: Ann. Phys. 326, 96 (2011), RMP 77, 259 (2005) Hallberg: Adv. Phys. 55, 477 (2006)

7 Infinite system DMRG Calculate the DM and reduce the dimension Add two sites at the center Iterate until the desired system size is reached

8 Finite system DMRG reduce the dimension to m prepared in previous steps we know the transfer matrix wavefunction in new basis can be predicted One of the blocks can be made longer iteratively

9 Finite system DMRG predicted wavefunction iteratively updated m can be increased during the process for higher precision Iterate until physical quantities (e.g. energy) converge

10 Application of DMRG: Low-dimensional quantum systems DMRG: variational method Error in ground state energy is positive, and decrease as # of states m is increased 1D Heisenberg model (White: PRB 48, (1993)) H Si Si 1 i Conjecture: gapped for spin, gapless for spin +1/2 (Haldane PRL 1983) S=1 Haldane gap detected ~ 150 papers per year! An improved method gives (4) (Ueda and Kusakabe: PRB 84, (2011)) Eric Jeckelmann, April 16, dmrg/paper_stat5.pdf

11 Fractional Quantum Hall systems (Landau levels: effectively 1D) Application of DMRG: Low-dimensional quantum systems Mixtures of bosons and fermions Spin-polarized half-filled fermions + bosons Mering and Fleischhauser: PRA 81, (R) (2010) Phase separation between CDW and Mott insulator Correlated electrons + phonons Shibata and Yoshioka: PRL 86, 5755 (2001) Phase diagram of Hubbard-Holstein model Tezuka, Arita and Aoki: PRB 76, (2007) Also, quantum chemistry, 2D & 3D classical systems,

12 Two-component Fermi gas eutral atoms: bosons or fermions depending on parity of A + Z (nucleon number + electron number) 6 Li Atom: fine structure, hyper fine structure electron spin S, orbital degrees of freedom, nuclear spin I Gupta et al.: Science 300, 1723 (2003) Fermions in two hyperfine states: Loss due to three-body collisions is rare Pauli principle (pseudo-) spin population preserved Fermi gas with fixed spin imbalance can be realized

13 Feshbach resonance A (highly excited) molecular state close to E B =0 can modify the scattering length Finite overlap of wavefunctions Closed channel: Quantum numbers differ from initial state of Rapid change of scattering length as function of magnetic field B energy When the open and close channels differ in magnetic moment, we can utilize closed channel open channel 1/(k F a)>0 internuclear distance Feshbach resonance: One bound (=molecular) state has binding energy = 0 1/(k F a)=0 energy closed channel 1/(k F a)<0 interaction (including sign) can be controlled open channel internuclear distance magnetic field B

14 For equal number of atoms, BEC-BCS crossover BEC : Bosons might break into fermions at energy Δ, but Δ is not correlated with T c BCS-type condensate : Pairing gap Δ is in proportion to T c (density determines T c ) BEC of diatomic molecules : smoothly connected with BCS condensate? Theory: Eagles (1969), Leggett (1980), ozières and Schmidt-Rink (1985), Experiment for the same number of Fermi atoms in two of the hyperfine states: 40 K BCS-like superfluid BEC of molecules Greiner et al.: PRL (2003, 2004) ature (2003) What happens when the numbers of two spins are not equal?

15 1) Harmonically trapped imbalanced system Motivation: condensation of population imbalanced fermions in elongated traps Zwierlein et al.(mit): Science 311, 492 (2006) Partridge et al.(rice): Science 311, 503 (2006) Discrepancy in cloud shape and maximum imbalance P for condensate More recent experiment: ascimbène et al. (ES): ature 463, 1057 (2010) P=( - )/( + ) Q. What happens when the atom trap is essentially 1D? 1 s-wave scatt. length ( g ) Kinetic energy << level separation of the radial trap Olshanii: PRL D optical lattice array of 1D traps realized (Esslinger group, ETH Zürich; Hulet group, Rice) I. Bloch: ature Phys. (2005)

16 3D: Controversy over experiments MIT Rice 5 Trap aspect ratio λ 50 ~10 7 umber of 6 Li atoms ~10 5 ~equipotential surface Density distibution significant deformation Radial dir. 1 1/α Axial dir. (Situation in ) Shin et al.:prl 97, (2006) Partridge et al.: PRL 97, (2006) P CC =75-80% Upper limit for imbalance P for condensation P=( - )/( + ) : majority : minority P CC >95% (two hyperfine states of 6 Li) What happens in 1D?

17 Pair with finite momentum: FF and LO states Fulde-Ferrell state Larkin-Ovchinnikov state Different Fermi momentum for up and down spins Pairs with non-zero momentum condense LO : density difference oscillates with wave number 2q Δ~exp(iqx) Δ~cos(qx) Density difference Order parameter Machida and akanishi: PRB 30, 122 (1984)

18 Confining potential ( r) Ar Discretization of the space Apply DMRG 2 U(r) = gδ(r) Kinetic energy 2 2 K( k) k / 2m Site potential i Aa 2 i L 1 2 L-site Hubbard model cf. optical lattice systems λ=2a 2V 0 2 a H t i, j U = g / a t 2 K / 2ma 2 K. E. dispersion 2 2 ( k) 2t (1 cos ka) ta k cˆ ˆ ˆ ˆ ˆ ˆ i c j c j ci U n n i i,, i i, nˆ i i

19 Density difference DMRG simulation of continuous system with the lattice introduced Smaller lattice spacingcontinuum limit approached 40 atoms L: number of sites ormalized real-space coordinate Density difference: shows oscillation incommensurate with lattice

20 up down difference Pair correlation and density distribution M. Tezuka and M. Ueda, PRL 100, (2008) Pair correlation j i j j i i z z c c c c ) ( 0,,,, ) ( 0 ˆ ˆ ˆ ˆ P imbalance parameter positive negative

21 up down difference Pair correlation and density distribution M. Tezuka and M. Ueda, PRL 100, (2008) Pair correlation j i j j i i z z c c c c ) ( 0,,,, ) ( 0 ˆ ˆ ˆ ˆ P imbalance parameter positive negative

22 up down difference Pair correlation and density distribution M. Tezuka and M. Ueda, PRL 100, (2008) Pair correlation j i j j i i z z c c c c ) ( 0,,,, ) ( 0 ˆ ˆ ˆ ˆ P imbalance parameter positive negative

23 up down difference Pair correlation and density distribution M. Tezuka and M. Ueda, PRL 100, (2008) Pair correlation j i j j i i z z c c c c ) ( 0,,,, ) ( 0 ˆ ˆ ˆ ˆ P imbalance parameter positive negative

24 up down difference Pair correlation and density distribution M. Tezuka and M. Ueda, PRL 100, (2008) Pair correlation j i j j i i z z c c c c ) ( 0,,,, ) ( 0 ˆ ˆ ˆ ˆ P imbalance parameter positive negative

25 up down difference Pair correlation and density distribution M. Tezuka and M. Ueda, PRL 100, (2008) Pair correlation j i j j i i z z c c c c ) ( 0,,,, ) ( 0 ˆ ˆ ˆ ˆ P imbalance parameter positive negative

26 up down difference Pair correlation and density distribution M. Tezuka and M. Ueda, PRL 100, (2008) Pair correlation j i j j i i z z c c c c ) ( 0,,,, ) ( 0 ˆ ˆ ˆ ˆ P imbalance parameter positive negative

27 Condensate? two-body density matrix Fermions in M states: Maximum possible eigenvalue = (M-+2)/M ~ (C.. Yang, RMP 1962) Measure of pair condensation Diagonalize to obtain eigenvalue distribution Eigenfunction: state occupied by (, ) pairs L 4 matrix elements for L-site chain cf. Condensate fraction for Bose gas one-body DM

28 Largest eigenvalue Eigenvalue Eigenvalue distribution Almost irrelevant of total number of atoms P Kinks observed on-int. atoms Eigenstate number Most of minority atoms contribute to quasi-condensate M. Tezuka and M. Ueda: PRL 100, (2008)

29 LO condensate at trap center Pair correlation: periodic sign change Population difference: constant + oscillation in 1D k n F M. Tezuka and M. Ueda: PRL 100, (2008) P=0.1 P=0.3

30 Phase diagram P=0.1 o up-only region + equal-population P=0.4 Up-only; Phase sep.

31 Trap depth Density 1D: LDA (local density approximation) results Exact solution for system without trap (Yang s generalied Bethe ansatz Gaudin s integral equation) Orso, PRL 98, (2007) Hu, Liu and Drummond, PRL 98, (2007) Small P Large P FFLO Chemical potential difference Consistent with our DMRG results

32 1D Experiment (Rice group) Density at central tube reconstructed Liao et al.: ature 467, 567 (Sep 2010) P=0.015 P=0.055 P=0.10 P=0.33 Majority Minority T/T F ~ 0.15 (fit to thermodynamic Bethe ansatz with LDA) Low enough for condensation? Difference P=0.05 P=0.10 P=0.35 Our data at T=0 (interaction strength OT tuned)

33 Attraction strength Conclusion: (1) Population imbalance + harmonic trap Pairing? LO (quasi-) condensate Phase separation? Yes (LO at center) Upper limit in imbalance P for condensation? not observed M. Tezuka and M. Ueda: see also: PRL 100, (2008); JP 12, (2010) Feiguin and Heidrich-Meisner: PRB 76, R (2007); PRL 102, (2009) (Ladder); Lüscher, oack, and Läuchli: PRA 78, (2008); Batrouni et al.: PRL 100, (2008) (QMC) Machida, Yamada, Okumura, Ohashi, and Matsumoto: PRA 77, (2008); Rizzi et al.: PRB 77, (2008); Machida et al.: PRB 78, (2008)

34 2) Optical lattice with disorder Disorder: enhance? suppress? Superconductivity Superfluidity Interaction: pairing force? other phase? Population imbalance = magnetic field : FFLO ow widely controllable in cold Fermi atomic gases Optical lattice laser standing wave Bichromatic lattice [Roati et al.: ature 453, 895 (2008)] Holographic potential imprinting in 2D M. Greiner s group *Gillen et al.: PRA 2009; Bakr et al.: ature 2009]

35 Experimental realization Optical lattice laser standing wave Bichromatic lattice G. Roati et al., ature 453, 895 (2008) (Firenze-Rome-Trento) Another possibility: holographic potential imprinting in 2D M. Greiner s group J. I. Gillen et al., PRA 80, (R) (2009) W.S. Bakr et al., ature 462, 74 (2009) Our motivation: what happens in 1D? Quantum fluctuation suppresses true long range order (even for T=0) Finite system : can have condensate (superfluid) Is coherence length O(system size)? Can be studied with numerically exact low-energy methods (Here we use DMRG)

36 Existing results Speckle potential (Gaussian random) All eigenstates exponentially localized L. Sanchez-Palencia et al.: PRL 98, (2007) A.M. García-García and E. Cuevas: PRB 79, (2009) Fibonacci potential ABAABABA... J. Vidal et al.: PRL 83, 3908 (1999), PRB 65, (2001); K. Hida: PRL 86, 1331 (2001) Critical irrespective of the strength of λ Bichromatic potential n cos n V 2 Aubry-Andre model non-interacting : metal-insulator transition at λ=2j (J: hopping) umerical studies of interacting systems Bose Hubbard (DMRG): Deng et al.: PRA 78, (2008); Roux et al.: PRA 78, (2008) Spinless Fermions : Chaves and Satija (ED, PRB 55, (1997)); Schuster et al. (PBC DMRG, PRB 65, (2002))

37 Quasiperiodic potential Hopping J Interaction U Laser standing waves J Disorder amplitude 2λ U 2λ Modeled by a single-band Hubbard model with site level modification

38 Formulation: Hubbard model + quasiperiodic potential J: unit of energy(=1) U: negative for attractive interaction n cos2n, F / F 5 1 Ratio of consective Fibonacci numbers golden ratio (=irrational number) as k ( ς, L) = (10,90), (26, 234), (42, 378) : ν=2/9 V F 0 F 1 1, F k2 F k1 F k k k1 / 2, L F k1 1 on-interacting case: all eigenstates become critial at λ=2j

39 One-electron level scheme (non-interacting) λ=1 λ=2 λ=3 site number i λ=10 λ=3 site number i site number i 611 sites; 1st, 10th, 306th, 611th eigenstate wavefunction Fermions localize for λ>2 for U =0 λ=1 (Energy spectrum is fractal and changes smoothly as λ is increased)

40 Interaction strength egative U: interaction energy linearly increase as U is increased Positive U: it has a peak because double occupancy is suppressed as Ularge t=1 (band width=4) U =6 : strongly interacting U =1 : weakly interacting band width

41 How to detect pairing and delocalization? Pairing On-site pair correlation function: easy to calculate with DMRG Depends on the site potentials of the site pair Averaged equal-time pair structure factor Sum of pair correlation for all lengths average over sites cf. Hurt et al.: PRB 72, (2005); Mondaini et al.: PRB 78, (2008) Delocalization i, r cˆ cˆ cˆ cˆ ir, ir, i, i, Phase sensitivity: requires (anti-)periodic condition [see e.g. Schuster et al.: PRB 65, (2002) ] Hard to calculate within DMRG (not open BC) in large systems (OK for small systems) 2 I ˆ ˆ E ni n 1, 1 i, Inverse participation ratio (IPR) i Add 2 atoms How uniformly is the population increase distributed? Compare between different system sizes s r P i, r Increasing function of L if decay of correlation is slow i 1

42 The case without disorder (λ=0) Pair structure factor indicator of global (quasi long-range) superfluidity Inverse participation ratio indicator of atom delocalization Both increase with U, and system size L

43 Pair structure factor Weak U Tezuka and García-García: PRA 82, (2010) U=-1 : Quasi long-range pairing disappears (λ p ~0.95) before localization (λ c ~1.00) Pairing enhanced by disorder Inverse participation ratio

44 Pair structure factor Larger U Tezuka and García-García: PRA 82, (2010) U=-6 : Quasi long-range pairing disappears at localization (λ c ~0.30) Inverse participation ratio

45 Delocalized, superconducting pairs below sc-metal transition Tezuka and Garcia-Garcia: PRA 82, (2010) Localized, tightly bound pairs at large λ Spin gap Minimum energy to break a pair by spin flipping n, n 1 E n n s E0 1 0, Continues to increase even after λ c

46 M. Tezuka and A. M. Garcia-Garcia: PRA 82, (2010) Disorder Schematic phase diagram Effect of coexisting disorder (bichromatic potential) and short-range attractive interaction Studied for 1D fermionic atoms on optical lattice For strong attraction ( U J), pairing decreases as disorder λ is increased, and localizes at ~ insulating transition λ c 2 λ/j Insulator, not superfluid spin, charge gaps For weaker attraction ( U ~J), pairing has a peak as a function of disorder λ, but localizes before λ c enhanced Global superfluidity Metal without superfluidity Attraction U /J

47 What happens for interacting fermions? What about dynamics? Many experiments observe the dynamics of the atomic clouds after release from a trap Bosons: E. Lucioni et al. (LES, Florence): PRL 106, (2011) Subdiffusion observed in bichromatic lattice (3D) V(x) = V 1 cos 2 (k 1 x) + V 2 cos 2 (k 2 x), k 1 =2π/(1064.4nm), k 2 =2π/(859.6nm) 50 thousand 39 K atoms, almost spherical trap switched off at t=0 Initially a=280a 0 (repulsive), λ~3j (localized) tuned to final value within 10 ms λ=4.9j α=0.5: normal diffusion width ς: square root of the second moment E int =1.8J E int =2.3J E int =0 J/h = 300Hz α=0.5: normal diffusion J/h = 180Hz Simulation (1D DLSE) Larcher et al. PRA 80, λ=5.3(4)j

48 Does the phase depend on filling? What do we see? If the phase diagram is sensitive to the filling Density decrease after release may induce (de)localization Intuitively, The density is decreased as the atoms flow to the outer side of the system; density of states at the Fermi surface changes (not monotonously) But we checked The ground state phase diagram: does not depend strongly up to filling ~ 0.31 (per spin per site) λ=10 λ=3 λ=1 Small change of density of states does not affect λ loc strongly Gap at ~31% filling

49 One parameter scaling theory Abrahams et al.: PRL 42, 673 (1979) see also Garcia-Garcia and Wang: PRL 100, (2008) L : system size, δ : (one-particle) mean level spacing Dimensionless conductance g(l) = E T / δ E T : Thouless energy ( related to typical time for particle to travel L ) <x 2 (t)> t α (d>2)d ormal metal : g(l) L d-2 E T L -2, δ L -d --- Metal-insulator transition : g(l) = g c --- Insulator : g(l) exp(-l/ξ) 0 Motion slowed down, ~ L 2/α time to propagate L, E T -1 t -2/α, α<1 MIT should occur at α = 2d H /d = 2d H (has been checked for the non-interacting case; Artuso et al.: PRL 68, 3826 (1992); Piechon et al.: PRL 76, 4372 (1996)) Multifractal spectrum with Hausdorff dimension d H δ L -d/d H Localization length ξ: should diverge as λ-λ c -ν (ν=1 at U=0)

50 ear metal insulator transition Disordered 1D system, U < 0 U 0 intermediate U U Diffusion <x 2 (t)> t α α~1 α~2 brownian motion? ballistic motion Hausdorff dimension d H ~0.5 One parameter scaling d H ~1 of the spectrum d H α = 2d H at MIT ot fractal see e.g. Artuso et al.: PRL 68, 3826 (1992) Localization length close to transition Our conjecture ν = 1/(2d H ) = 1/α? ξ λ λ c -ν mean-field like; similar to Cayley tree Let us numerically check by studying the dynamics

51 Setup (1) Optical lattice + incommensurate potential (Aubry-André model) On-site attractive interaction Initially trapped in a harmonic potential Remove the harmonic potential but keep the incommensurate potential on: what happens? Study by time-dependent DMRG

52 Finite system DMRG wavefunction constructed from the previous step Iterate until physical quantities (e.g. energy) converge

53 Time-dependent DMRG White and Feiguin: PRL 93, (2004) Application of exp(-iτh i,j ) is almost exact if H i,j only affects neighboring sites i, j T/τ finite system iterations to reach time T Hˆ Hˆ 1,2 exp( ihˆ ) Suzuki-Trotter decompotision exp( ihˆ exp( ihˆ exp( ihˆ Hˆ 2,3 2,3 L / 2, L / 21 Hˆ / 2) exp( ihˆ / 2) exp( ihˆ L2, L1 L2, L1 / 2) exp( ihˆ Hˆ L2, L1 L1, L 2,3 L / 2, L / 21 / 2) exp( ihˆ / 2) exp( ihˆ 3 / 2) O( ) L1, L 1,2 ) ) exp( ihˆ L / 2, L/ 21 / 2) exp( ihˆ L / 21, L/ 22 / 2) Operator applied to the wavefunction at each step exp( ihˆ L, exp( ihˆ L 2, L1 / 2) 1 L ) exp( ihˆ 1, 2) exp( ihˆ 2, 3 / 2) exp( ihˆ L / 2, L/ 21 / 2)

54 Time-dependent DMRG: other schemes Cazalilla and Marston: PRL 88, (2002) Luo et al.: PRL 91, (2003) Vidal: PRL 93, (2004) Feiguin and White: PRB 72, (2005) Dutta and Ramasesha: PRB 82, (2010) Time-evolving block decimation (TEBD) has also been applied to cold atom systems e.g. Macroscopic quantum tunneling between different supercurrent states in a ring Danshita and Polkovnikov: PRB 82, (2010)

55 Applications of time-dependent DMRG on cold atom systems Propagating 1D density waves of bosons in optical lattice Kollath et al.: PRA 71, (2005) Spin-charge separation in S=1/2 Hubbard model Kollath et al.: PRL 95, (2005) Spin-charge separation in two-component Bose- Hubbard model Kleine et al.: PRA 77, (2008)

56 Applications of time-dependent DMRG on cold atom systems Andreev-like reflection in bosons Daley et al.: PRL 100, (2008) Optical superlattice quenched Yamamoto et al.: JPSJ 78, (2009)

57 Setup (1) Optical lattice + incommensurate potential (Aubry-André model) On-site attractive interaction Initially trapped in a harmonic potential Remove the harmonic potential but keep the incommensurate potential on: what happens? Study by time-dep. DMRG for Hubbard model

58 on-interacting case Smooth diffusion until bouncing back at (artificial) system boundary

59 λ=0.5 < λ c (U=-1) Atoms quickly flow to both sides

60 λ=1.0 ~ λ c (U=-1) Significantly slower diffusion

61 λ=1.5 > λ c (U=-1) Atoms move only locally

62 Setup (2) Optical lattice + incommensurate potential (Aubry-André model) On-site attractive interaction Initially trapped in a box potential Remove the box potential but keep the incommensurate potential on: what happens? Study by time-dep. DMRG for Hubbard model

63 Energy change Energy change Check: energy is nearly preserved 160 sites, fermions, Δt = 0.01 U=-1 U= t=100 (10 4 steps) t=100

64 Suppressed motion for λ > λ c 200 sites, 18 up and 18 down fermions, U=-1 λ=0.6 λ=0.9 λ c ~ 1.0 λ=1.1

65 J. Ozaki, M. Tezuka, and. Kawakami: arxiv: ) Collision dynamics Elongated two-component Fermi gas: up and down spins released from separate traps to collide ( Little Fermi Collider ) cf. 1D Bosons: absence of thermalization Kinoshita et al.: ature 440, 900 (2006) Spin diffusion constant minimum at Feshbach resonance; converges as T0 Universal spin transport in a strongly interacting Fermi gas Sommer et al.: ature (2011) Motion of center of mass 1sec What happens at 1D?

66 Motivation What kind of many-body effects are observed during a single collision between two one-dimensional fermion clusters? A spin-dependent harmonic trap Quenched to a shared potential The fermions collide at the trap center Model: Hubbard model Method: time-dependent DMRG

67 Example U/J = 0.80 (7+7 atoms, ~55% reflectance)

68 Weak interaction: most of atoms are not reflected particle reflectance for n + n atoms R n : R1 u 2 (u 0) Quasi-classical model Quasi-classical model: a series of one-to-one collisions between two types of independent classical particles (i) u 0 : n 2 times of independent spin-up and spin-down collisions R qc n = n 2 R1/n = nr1 Consistent with the simulation

69 Strong interaction: most of atoms are reflected back Transmittance for n + n atoms Tn : T1 u -2 (u ), Rn + Tn = 1 Quasi-classical model Quasi-classical model: a series of one-to-one collisions between two types of independent classical particles (ii) u : n atoms collide successively against n atoms T qc n = nt1/n = T1 Inconsistent with the simulation! What is going on? The work is in progress.

70 Summary Application of DMRG for static and dynamic behavior of Fermi cold atom gases in 1D Population-imbalanced gas in harmonic trap: FFLO-like condensate Quasiperiodic disorder PRL 100, (2008); ew J. Phys 12, (2010) Can enhance condensation for weak attraction Trap-release dynamics close to metal-insulator transition: anomalous diffusion observed PRA 82, (2010) arxiv: Collision of spin clusters: more atoms pass through than quasi-classically expected = emergent many-body behavior There is much more to explore with powerful numerical methods!